Mathematics of Physics and Engineering

Scalar Triple Product 23

where PiPj = OPj — OPi- If (xi, yi, Zi) are the cartesian coordinates of the
point Pi, then the criterion for collinearity (1.2.28) becomes

det

Z J K

X 2 ~ Xl 2/2 - 2/1 Z 2 - Z\

Xj, - Xl 1/3 - 2/1 23 - Z\

= 0. (1.2.29)

In the following three exercises, the reader will see how the mathematics

of vector algebra can be used to solve problems in physics.

EXERCISE 1.2.12.C Suppose two forces F\, F 2 are applied at P; r = OP.

Show that the total torque at P is T = Ti + T 2 , where Ti = r x Fi and

T 2 = r xF 2.

EXERCISE 1.2.13/^1 Consider a rigid rod with one end fixed at the origin O

but free to rotate in any direction around O (say by means of a ball joint).

Denote by P the other end of the rod; r = OP. Suppose a force F is applied

at the point P. The rod will tend to rotate around O.

(a) Let r = 2i + 3j + k and F = i + j + k. Compute the torque T. (b)

Let r = 2 i + Aj and F = i + j, so that the rotation is in the (i, j) plane.

Compute T. In which direction will the rod start to rotate?

EXERCISE 1.2.14/^4 Suppose a rigid rod is placed in the (i, j) plane so that

the mid-point of the rod is at the origin O, and the two ends P and Pi have

position vectors r = i + 2 j and T"i = — i — 2 j. Suppose the rod is free to

rotate around O in the (i, j) plane. Let F = i + j and Fi = — i — j be two

forces applied at P and Pi, respectively. Compute the total torque around

O. In which direction will the rod start to rotate?

1.2.3 Scalar Triple Product

The scalar triple product (u,v,w) of three vectors is defined by

(u, v, w) = u • (v x w).

Using (1.2.24) it is easy to see that, in cartesian coordinates,

(u, v, w) = det

Ul «2 ""3

Vi V 2 V3

Wi W 2 U>3

From the properties of determinants it follows that

(u,v,w) = —(v,u,w) = (v,w,u) = (w,u,v)